Biplanar homogeneous field electromagnets and method for making same

ABSTRACT

Biplanar, symmetrical electromagnets for providing a homogeneous magnetic field. The magnets have coils disposed in two parallel planes. The coils in the two planes are identical. The radii and Ampere-turns of the coils are selected so that a magnetic field between the planes is homogeneous. One preferred embodiment has 6 coils, with 3 coils in each plane. Other embodiments have 8, 10, 12, or more coils. The method of making the coils begins with an equation for the spherical harmonic coefficients describing the fields from a coil as a function of radius, position and Ampere-turns. For a magnet with K coils, the first K-1 even spherical harmonic coefficients are set equal to zero (the odd coefficients are zero due to symmetry of the magnet). This produces a set of equations that, when solved, provides the radii, positions, and Ampere-turns of the K coils. The method can be used to design a biplanar, symmetrical electromagnet with any even number of coils.

RELATED APPLICATIONS

This application claims priority from provisional patent application60/081,311 filed on Apr. 10, 1998, which is incorporated herein byreference.

FIELD OF THE INVENTION

This invention relates generally to electromagnets. More particularly,it relates to methods for designing homogeneous field electromagnets foruse in magnetic resonance imaging (MRI) or prepolarized magneticresonance imaging (PMRI).

BACKGROUND OF THE INVENTION

Magnetic resonance imaging (MRI) is a common and well known techniquefor imaging the internal structure of objects and for medical diagnosis.Conventional MRI requires that the object to be imaged be placed in ahomogeneous (typically to within 40 ppm) and strong (typically in therange of 0.5 to 1.5 Tesla) magnetic field. Generating such magneticfields is difficult and expensive.

Prepolarized MRI (PMRI) is a recent technique which uses a strong,nonhomogeneous pulsed magnetic field in combination with a weaker,homogeneous magnetic field to perform imaging. The strong, pulsed fieldis known as the polarizing field and it is produced by a polarizingmagnet. The weaker, homogeneous field is known as the readout field andis produced by a readout magnet. PMRI is also referred to asswitched-field MRI and is related to field cycling nuclear magneticresonance (NMR) relaxometry.

In PMRI, the polarizing field is switched on briefly (about 0.01 to 2seconds) to polarize the nuclear spins inside the object to be imaged.Then, the polarizing field is rapidly reduced at a rate faster than thedecay rate of the nuclear spin polarization. The nuclear spinpolarization is then analyzed in the readout magnetic field. Thepolarizing field causes the nuclear spin polarization to be greater thanit would be with only the readout field. Reference can be made to U.S.Pat. Nos. 5,629,624 to Carlson et al., 4,906,931 to Sepponen, and5,057,776 to Macovski concerning PMRI.

A result of the pulsed polarizing magnetic field is that it renders alarge PMRI device very difficult to build. The magnetic energy stored inthe magnet must be removed and restored with every pulse. Thispractically limits the amount of energy which can be stored in thepulsed magnet and thus the size of the PMRI device. Therefore, futurePMRI devices will likely be small dedicated imagers, dedicated toimaging small body parts such as hands, feet, knees, heads, breasts,neck and the like.

Imaging small body parts places limitations on magnet geometry (bothreadout and polarizing). Most body parts are not cylindrical andtherefore do not efficiently occupy the volume inside a traditionalcylindrical magnet assembly. A cylindrical magnet assembly is acollection of coils arranged on the surface of a cylinder. Access to themagnetic field of a cylindrical magnet is limited to the end openings ofthe cylinder or between the coils. This limited access makes itdifficult and uncomfortable to image certain body parts such as knees.Of particular difficulty is providing a magnet for imaging a knee orelbow as it is flexing.

It would be an advance in the art to provide readout magnet designswhich allow increased access to the homogeneous magnetic field. Suchimproved magnet designs would be particularly well suited for use indedicated PMRI machines.

Lee-Whiting discloses a design for a 4-coil biplanar magnet in“Homogeneous Magnetic Fields”, Atomic Energy Commission of CanadaLimited CRT-673, 1-29 (1957). The magnet has 2 coils symmetrically andcoaxially disposed in each of two parallel planes. The homogeneousmagnetic field is located between the planes defined by the coils. Thehomogeneous magnetic field can be accessed from the radial direction(i.e., from between the planes defined by the coils). A similar 4-coildesign is also disclosed by Garrett in “Thick Cylindrical Coil Systemswith Field or Gradient Homogeneities of the 6th to 20th Order”, Journalof Applied Physics, 38, 2563-2586 (1967). A shortcoming of these 4-coildesigns is that they are relatively inefficient in producing the desiredhomogeneous field, and produce relatively inhomogeneous magnetic fields.

U.S. Pat. No. 4,829,252 to Kaufman discloses an MRI system with improvedpatient access to the magnetic field. The system of Kaufmann usesbiplanar magnets to produce the required homogeneous magnetic field.Kaufman does not disclose how to design the biplanar magnets or specificbiplanar magnet designs.

OBJECTS AND ADVANTAGES OF THE INVENTION

Accordingly, it is a primary object of the present invention to providea method for designing magnets that:

1) produces biplanar magnet designs with improved magnetic field accesscompared to prior art designs;

2) produces biplanar magnet designs with exceptional magnetic fieldhomogeneity;

3) can be used to design biplanar magnets with relatively large numberof coils.

It is a further object of the present invention to provide biplanarmagnets that:

1) produce exceptionally homogeneous magnetic fields;

2) are relatively efficient;

3) have improved access to the homogeneous magnetic field.

It is also an object of the present invention to provide an apparatusfor prepolarized magnetic resonance imaging (PMRI) that:

1) has improved access to the imaging region;

2) can be adapted to image many different body parts.

These and other objects and advantages will be apparent upon reading thefollowing description and accompanying drawings.

SUMMARY OF THE INVENTION

These objects and advantages are attained by a 6-coil biplanarsymmetrical electromagnet. The 6-coil magnet has 3 coils symmetricallyand coaxially disposed in each of two parallel planes. The coils haveradii and are designed to carry accurately determined currents (inunites of Ampere-turns). The 6-coil magnet provides an accuratelyhomogeneous magnetic field between the two planes. The coils enclose anideal filamentary current loop calculated according to a method of thepresent invention. The magnet may further include electronics forproviding the accurately controlled currents for the coils. The presentinvention also includes 8-coil and 10-coil biplanar symmetricalelectromagnets.

The magnet may also include a polarizing magnet for providing apolarizing magnetic field needed for performing prepolarized magneticresonance imaging (PMRI). The polarizing magnet can be oriented in manydifferent ways.

The present invention also includes a method for making biplanarsymmetrical electromagnets having K coils. The method begins withproducing equations for the spherical harmonic coefficients describingthe magnetic fields produced by the K coils. The equations for thespherical harmonic coefficients are set equal to zero, which then allowsa numerical optimization program to solve the equations for the coilradii, coil locations, and coil currents in units of Ampere-turns. Thenumber of coils K can be equal to or greater than 6.

The present invention also includes magnets made according to the methodof the present invention.

Also, the present invention includes an apparatus for performingprepolarized magnetic resonance imaging (PMRI). The apparatus has abiplanar, symmetrical readout electromagnet made according to thepresent invention and a polarizing magnet. The readout electromagnetprovides a homogeneous magnetic field. The polarizing magnet can havemany different orientations with respect to the readout electromagnet.

DESCRIPTION OF THE FIGURES

FIG. 1 shows a 6-coil biplanar, symmetrical electromagnet according tothe present invention.

FIG. 2 shows an edge-on view of the electromagnet of FIG. 1.

FIG. 3a shows a side view of the electromagnet of FIG. 1.

FIG. 3b shows a cross sectional view of the magnet of FIG. 1.

FIG. 4 shows a plot of field homogeneity contours of the magnet of FIG.1.

FIG. 5 illustrates variables for describing coils in the electromagnetsof the present invention.

FIGS. 6-9 show different ways in which the biplanar, symmetrical magnetscan be used in a prepolarized magnetic resonance imaging (PMRI) device.

FIG. 10 shows a conventional magnetic resonance imaging device accordingto the present invention.

DETAILED DESCRIPTION

The present invention provides a method for designing electromagnetswith an exceptionally homogeneous magnetic field and improved access tothe homogeneous magnetic field. All magnets designed according to thepresent invention are symmetrical and biplanar.

FIG. 1 shows a 6-coil magnet designed according to the method of thepresent invention. The magnet is biplanar, and has a small coil 20 a, amedium coil 20 b, and a large coil 20 c located in a first plane 26; andhas identical coils 22 a, 22 b, 22 c located in a second plane 28. Firstplane 26 and second plane 28 are parallel and spaced apart by a distance2z₁ 30. All the coils are coaxial about axis 27.

FIG. 2 shows an edge-on view of the magnet of FIG. 1. The distance 2z₁30 between planes 26, 28 is clearly shown. The magnet is symmetricalacross symmetry plane 24. Coils have widths 31 a, 31 b which arepreferably centered on the first plane 26 and second plane 28,respectively.

FIG. 3a shows a side view of the magnet of FIG. 1. The coils 20 a-c, 22a-c have accurately selected radii 32, 34, 36. The radii of the coils isexpressed as a function of the distance 2z₁ 30. The radii are defined asthe distance from the axis 27 to the center of each coil. The followingtable presents the radii of the coils as a function of the distance 2z₁.

TABLE 1 Coil Radius Small Coils 20a, 22a 0.394771z₁ Medium Coils 20b,22b 0.858722z₁ Large Coils 20c, 22c 2.491120z₁

In operation, the coils carry accurately controlled currents. Thecurrents carried by the coils are given in the table below.

TABLE 2 Coil Ampere-Turns (normalized) Small Coils 20a, 22a 1I_(s)Medium Coils 20b, 22b 3.596420I_(s) Large Coils 20c, 22c 118.3143I_(s)

Increasing or decreasing the currents in the coils while maintaining thenormalized current values will only change the magnetic field magnitude.The magnetic field homogeneity and distribution will be preserved. TheAmpere-turns value I_(s) is arbitrary.

FIG. 3b shows a cross sectional view of the magnet of FIG. 1. The radiivalues given in Table 1 correspond approximately to the radii of thecenters of the coils 20 a-c and 22 a-c. However, it is understood thatthis is not a necessary feature of the present invention.

The magnet of FIG. 1 having the given coil radii and carrying the givencurrents provides an accurately homogeneous axial magnetic field in thecenter of the magnet between the planes 26, 28. FIG. 4 shows a plot ofthe magnetic field distribution in a plane perpendicular to planes 26,28 and coincident with the magnet axis 27. 2z₁ is the distance 30between first plane 26 and second plane 28. Region 40 illustratesapproximately the volume where the magnetic field is homogeneous towithin 1 part-per-million. The magnetic field homogeneity is worse than10 parts-per-million outside region 42, and the magnetic fieldhomogeneity is worse than 100 parts-per-million outside region 44.

It is noted that the homogeneity of the magnetic field within region 40is sensitively dependent upon the positioning of the coils 20 a-c, 22a-c and the normalized magnitudes of the coil currents. Smalldisplacements of the coils causes relatively large changes in themagnetic field homogeneity; small changes in the coil currents alsocause relatively large changes in the magnetic field homogeneity.Therefore, it is important for the magnet of the present invention tohave coil positions and normalized coil currents that are accuratelyequal to the values given in Tables 1 and 2. However, small changes fromthe given values are within the scope of the present invention.Generally, the coil radii and normalized coil currents must be within 1%of the values given in Tables 1 and 2.

It is understood that radii and normalized current values close to thevalues given in Tables 1 and 2 will provide the most homogeneousmagnetic fields.

DESIGN METHOD

The present invention includes a method for designing biplanar,symmetrical electromagnets. The magnet of FIGS. 1-3 is a specificexample of a biplanar symmetrical magnet which has 6 coils. The methodof the present invention can be used to design biplanar, symmetricalelectromagnets with any even number of coils. For example, the presentmethod can be used to design biplanar symmetrical electromagnets with 8,10, 12, 14, or 16 coils.

FIG. 5 shows a geometry used to describe the current loops. Shown aretwo identical loops 46, 48 symmetrically disposed about an origin 50.

Generally, the position of the k^(th) loop is denoted by the sphericalcoordinates (R_(k), θ_(k)) and the current carried by the k^(th) loop isdenoted by I_(k) (in units of Ampere-turns). The radius of the k^(th)loop is denoted by a_(k), and the Z-position of the k^(th) loop is givenby z_(k). The radius a_(k) and Z-position z_(k) are related to thespherical coordinates by the transformation:

a _(k) =R _(k) sinθ_(k)

z _(k) =R _(k) cosθ_(k)

Using this geometry, the radial and axial magnetic field components at apoint (R,θ) can be written as: $\begin{matrix}{{B_{r}\left( {R,\theta} \right)} = {\frac{\mu}{2}\quad {\sum\limits_{n = 1}^{\infty}\quad {\frac{C_{n}}{n + 1}\quad R^{n}{P_{n}^{1}\left( {\cos \quad \theta} \right)}}}}} \\{{B_{z}\left( {R,\theta} \right)} = {\frac{\mu}{2}\quad {\sum\limits_{n = 0}^{\infty}\quad {C_{n}\quad R^{n}{{P_{n}\left( {\cos \quad \theta} \right)}.}}}}}\end{matrix}$

Where P_(n) are associated Legendre polynomials of the first kind anddegree (n) and order 1, and μ is the permeability of free space. Thevalues C_(n) are spherical harmonic coefficients and are given by:$\begin{matrix}{C_{n} = {- {\sum\limits_{k}{I_{k}R_{k}^{- {({n + 1})}}\sin \quad \theta_{k}{{P_{n + 1}^{1}\left( {\cos \quad \theta_{k}} \right)}.}}}}} & {{EQ}\quad 1}\end{matrix}$

Where P¹ _(n+1) are associated Legendre polynomials of the first kindand degree (n+1) and order 1. The coefficient for n=0 corresponds to thespatially invariant (DC) magnetic field component. Due to the magnetsymmetry, the odd-index C_(n) coefficients are zero (e.g., C₁=C₃=C₅ . .. 0).

Given that the magnet design is biplanar and symmetrical (necessaryconstraints in the present invention), the number of free designvariables are limited to the following:

Currents: (K/2−1) degrees of freedom. Since only current ratios affectthe field homogeneity, one of the coil currents is arbitrarily set toone.

Radii: (K/2−1) degrees of freedom. Since only coil radius ratios affectfield homogeneity, one of the coil radii is arbitrarily set to one.

Spacing between planes: One degree of freedom. The spacing can beexpressed in terms of the arbitrarily set radius, or vice versa.

Therefore, there exist (K/2−1)+(K/2−1)+1=K−1 different free designvariables for a biplanar, symmetrical magnet with K coils.

In the present method, the coil radii a_(k), coil currents I_(k), andspacing between planes 2z₁ (e.g., planes 26 and 28) are chosen so thatthe first K−1 even-indexed coefficients are zero. Therefore, thelowest-index non-zero coefficient has index N=2K. The first N-1coefficients are zero (i.e., C₁=C₂=C₃= . . . =C_(N−1)=0, and C_(N)≢0).The integer N is called the order of the magnetic field. Since bothradial and axial components of the field vary as R^(N), the magneticfield becomes more homogeneous in the vicinity of the origin as thefield order N increases.

The appropriate coil radii a_(k), coil currents I_(k), and spacing 2z₁are determined by setting the coefficients C_(n) equal to zero for evenvalues of n less than 2K. The resultant set of equations is then solvedfor a_(k) and 2z_(k) (both of which are expressed in terms of angleθ_(k) and R_(k)), and currents I_(k). Since the equations are nonlinear,they can be solved numerically on a computer having software such asMATLAB.

As a specific example, for a magnet with 6 coils, the followingequations are produced: $\begin{matrix}\begin{matrix}\begin{matrix}{0 = {\sum\limits_{k = 1}^{6}{I_{k}R_{k}^{- {({2 + 1})}}\sin \quad \theta_{k}{P_{2 + 1}^{1}\left( {\cos \quad \theta_{k}} \right)}}}} \\{0 = {\sum\limits_{k = 1}^{6}{I_{k}R_{k}^{- {({4 + 1})}}\sin \quad \theta_{k}{P_{4 + 1}^{1}\left( {\cos \quad \theta_{k}} \right)}}}}\end{matrix} \\{0 = {\sum\limits_{k = 1}^{6}{I_{k}R_{k}^{- {({6 + 1})}}\sin \quad \theta_{k}{P_{6 + 1}^{1}\left( {\cos \quad \theta_{k}} \right)}}}}\end{matrix} \\{0 = {\sum\limits_{k = 1}^{6}{I_{k}R_{k}^{- {({8 + 1})}}\sin \quad \theta_{k}{P_{8 + 1}^{1}\left( {\cos \quad \theta_{k}} \right)}}}} \\{0 = {\sum\limits_{k = 1}^{6}{I_{k}R_{k}^{- {({10 + 1})}}\sin \quad \theta_{k}{P_{10 + 1}^{1}\left( {\cos \quad \theta_{k}} \right)}}}}\end{matrix}$

When solved numerically for I_(k), R_(k), and θ_(k), this set of couplednonlinear equations provides the current and radius values given inTables 1 and 2 (where the radius values are expressed in terms of thespacing z₁) The 6-coil magnet of the present invention has a magneticfield order of 12.

The present invention can be applied to cases where the number of coilsK is any even number. For example, the present invention can be used todesign biplanar, symmetrical magnets with 8, 10, 12, 14, or more coils.

Electromagnets made according to the present invention are particularlywell suited for use as a readout magnet in prepolarized magneticresonance imaging (PMRI). PMRI requires a polarizing magnet in additionto a readout magnet. The magnets of the present invention offer severalpossibilities for conveniently locating the polarizing magnet.

FIG. 6 shows an example of a combination of a polarizing magnet andbiplanar readout magnet according to the present invention. FIG. 6 is anedge-on view of the first plane 26 and second plane 28. A solenoidalpolarizing magnet 60 provides a polarizing magnetic field 62 which isperpendicular to axis 27. Of course, a readout magnetic field providedby the biplanar coils in planes 26, 28 is parallel with the axis 27. Itis noted that the polarizing magnet 60 and coils of the biplanar magnetmust be sized so that they do not intersect.

FIG. 7 shows another example of a polarizing magnet combined with abiplanar readout magnet for use in PMRI. Two solenoidal polarizingmagnets 64 a, 64 b provide a polarizing magnetic field 66 parallel withaxis 27. The polarizing magnets are coaxial with coils of the biplanarreadout magnet (i.e., coaxial about axis 27).

FIG. 8 shows another example where a solenoidal polarizing magnet 68 isdisposed between planes 26, 28. The polarizing magnet 68 provides apolarizing field 70 perpendicular to the page.

FIG. 9 shows another example where a solenoidal polarizing magnet 72 isdisposed between planes 26, 28 and coaxial with the biplanar magnet axis27. The polarizing magnet provides polarizing magnetic field 74 forPMRI. In this embodiment, access to the homogeneous field is providedalong the axis 27, through plane 26 or through plane 28.

PMRI systems according to FIGS. 6-9 provide relatively open access tothe region having both the homogeneous field and polarizing field, whereimaging is provided.

Magnets according to the present invention can also be used inconventional magnetic resonance imaging (MRI) devices. FIG. 10 shows aconventional MRI device using a magnet according to the presentinvention. The device has a biplanar symmetrical magnet with coilsdisposed in first plane 26 and second plane 28. The biplanar symmetricalmagnet provides a homogeneous magnetic field necessary for MRI. Thedevice also has gradient coils 78, radiofrequency (RF) coils 80 forexciting nuclear spins, and a computerized console 82 for controllingthe gradient coils 78 and RF coils 80.

The present invention includes designs for 8 coil and 10 coil biplanar,symmetrical electromagnets. Given below are design parameters for thesemagnets. All the radius values are expressed in terms of z (one-half thespacing between coil planes), and all the Ampere-turn values arenormalized so that the current in the smallest coil is 1.

8-Coil Biplanar Design Parameters Ampere-Turns Coil Radius (normalized)Small 0.294922z 1I_(s) Medium 0.590474z 2.581089I_(s) Large 1.031283z7.938104I_(s) Extra large 2.632601z 233.1785I_(s)

There is a small coil, a medium coil, a large coil and an extra largecoil in each plane. All the coils are coaxial.

10-Coil Biplanar Design Parameters Ampere-Turns Coil Radius (normalized)Extra Small 0.236199z 1I_(s) Small 0.457269z 2.255341I_(s) Medium0.734885z 4.938906I_(s) Large 1.164685z 14.159399I_(s) Extra Large2.745650z 386.32597I_(s)

There is an extra small coil, a small coil, a medium coil, a large coiland an extra large coil in each plane. All the coils are coaxial.

Magnets according to the present invention are characterized in that thecoil radii, the coil currents, and the spacing between planes areprovided by numerical optimization of the nonlinear equations givenabove. An electromagnet according to the present invention will haveradii and currents specified by the above-described numericaloptimization procedure. The method generally provides a single globalsolution to the set of nonlinear equations used in the presentinvention.

It is understood that the radii given for 6-coil, 8-coil, and 10-coilmagnets are for ideal filamentary current loops with infinite currentdensity. When building a magnet according to the present invention, suchideal filamentary current loops must be approximated by real coilshaving a finite thickness and finite current density. Typically,filamentary loops are approximated by locating the centers of the realcoils at the same location as the ideal filamentary loops. Also,shimming techniques can be used to alter slightly the dimensions of thecoils or currents used in the coils. Given ideal filamentary looplocations and currents, a technician skilled in the art of magnet designand construction will be able to construct a real magnet providingessentially the same magnetic field as the ideal filamentary loops.

The present invention is equally applicable to making resistive magnetsand superconducting magnets. The magnets of the present invention can beeither resistive (e.g., copper coils) or superconducting.

It will be clear to one skilled in the art that the above embodiment maybe altered in many ways without departing from the scope of theinvention. Accordingly, the scope of the invention should be determinedby the following claims and their legal equivalents.

What is claimed is:
 1. A biplanar, symmetrical 6-coil electromagnetcomprising: a) a first small coil; b) a first medium coil; c) a firstlarge coil, wherein the first small coil, the first medium coil and thefirst large coil are located in a first plane; d) a second small coil;e) a second medium coil; and f) a second large coil, wherein the secondsmall coil, the second medium coil, and the second large coil arelocated in a second plane; wherein: 1) the first plane and the secondplane are parallel and spaced apart by a distance 2z; 2) all 6 coils arecoaxial about an axis; 3) the first small coil encloses a first smallfilamentary loop with a radius equal to 0.394771z, and the second smallcoil encloses a second small filamentary loop with a radius equal to0.394771z; 4) the first medium coil encloses a first medium filamentaryloop with a radius equal to 0.858722z, and the second medium coilencloses a second medium filamentary loop with a radius equal to0.858722z; 5) the first large coil encloses a first large filamentaryloop with a radius equal to 2.491120z, and the second large coilencloses a second large filamentary loop with a radius equal to2.491120z; 6) the first filamentary loops are coplanar with the firstplane, and the second filamentary loops are coplanar with the secondplane; whereby the electromagnet provides a homogeneous magnetic fieldbetween the first plane and second plane near the axis when energizedwith appropriate currents.
 2. The electromagnet of claim 1 furthercomprising a means for energizing the coils with electrical currentssuch that: a) the first small coil and the second small coil each carryAmpere-turns equal to an arbitrary value I_(s); b) the first medium coiland the second medium coil each carry Ampere-turns equal to3.596420I_(s) to within 1%; and c) the first large coil and the secondlarge coil each carry Ampere-turns equal to 118.3143I_(s) to within 1%.3. The electromagnet of claim 1 further comprising a polarizing magnetfor prepolarized magnetic resonance imaging, wherein the polarizingmagnet is disposed such that a polarizing magnetic field is provided inthe region of the homogeneous magnetic field.
 4. The electromagnet ofclaim 3 wherein the polarizing magnet is a solenoidal magnet disposedbetween the first plane and the second plane such that the polarizingmagnetic field is perpendicular to the homogeneous magnetic field. 5.The electromagnet of claim 3 wherein the polarizing magnet is asolenoidal magnet disposed between the first plane and the second planesuch that the polarizing magnetic field is parallel to the homogeneousmagnetic field.
 6. The electromagnet of claim 3 wherein the polarizingmagnet is a solenoidal magnet disposed outside the first plane andsecond plane and wherein the polarizing magnet is coaxial with the 6coils.
 7. The electromagnet of claim 1 wherein the coils are centeredupon the corresponding filamentary loops.
 8. The electromagnet of claim1 wherein each coil has cross-sectional dimensions less than 10% of thecoil's radius.
 9. A method for making a biplanar, symmetricalelectromagnet having K coils, where K is an even integer, comprising thesteps of: a) producing K-1 equations of the form$0 = {\sum\limits_{k = 1}^{K}{I_{k}R_{k}^{- {({n + 1})}}\sin \quad \theta_{k}{P_{n + 1}^{1}\left( {\cos \quad \theta_{k}} \right)}}}$

 wherein each equation has a different value of n, wherein each value ofn is a positive, even integer less than 2K, and wherein I_(k) denotesthe Ampere-tums and R_(k) and θ_(k) denote the spherical coordinates ofa k^(th) coil; b) solving the equations produced in step (a) for valuesof I_(k), R_(k), and θ_(k), wherein the coils are constrained to twoparallel planes and the coils are symmetrical; and c) building coilscorresponding to values of R_(k) and θ_(k) found in step (b) and capableof carrying Ampere-turns proportionate to values of I_(k) found in step(b).
 10. The method of claim 9 wherein K is equal to or greater than 6.11. The method of claim 9 wherein step (b) is performed numerically by acomputer.
 12. A biplanar, symmetrical electromagnet made according tothe method of claim 9 and having at least 6 coils.
 13. An apparatus forprepolarized magnetic resonance imaging, comprising: a) a biplanar,symmetrical readout electromagnet for providing a homogeneous magneticfield, wherein the readout magnet has a first plane and a second plane,wherein the readout magnet has at least 6 coils, and wherein the readoutmagnet is made according to the method of claim 8; and b) a polarizingmagnet disposed such that a polarizing magnetic field is provided in theregion of the homogeneous magnetic field.
 14. The apparatus of claim 13wherein the polarizing magnet is a solenoidal magnet disposed betweenthe first plane and the second plane such that the polarizing magneticfield is perpendicular to the homogeneous magnetic field.
 15. Theapparatus of claim 13 wherein the polarizing magnet is a solenoidalmagnet disposed between the first plane and the second plane such thatthe polarizing magnetic field is parallel to the homogeneous magneticfield.
 16. The apparatus of claim 13 wherein the polarizing magnet is asolenoidal magnet disposed outside the first plane and second plane andwherein the polarizing magnet is coaxial with the readout magnet.
 17. Anapparatus for magnetic resonance imaging, comprising: a) a biplanar,symmetrical electromagnet made according to the method of claim 8 andhaving at least 6 coils for providing a homogeneous magnetic field; b) agradient coil for providing gradient fields in the region of thehomogeneous magnetic field; c) a radiofrequency coil for excitingnuclear spins in the region of the homogeneous magnetic field; d) acomputerized console for controlling the gradient coil andradiofrequency coil.
 18. A biplanar, symmetrical 8-coil electromagnetcomprising: a) a first small coil; b) a first medium coil; c) a firstlarge coil; d) a first extra large coil, wherein the first small coil,the first medium coil, the first large coil, and the first extra largecoil are located in a first plane; e) a second small coil; f) a secondmedium coil; g) a second large coil; and h) a second extra large coil,wherein the second small coil, the second medium coil, the second largecoil, and the second extra large coil are located in a second plane;wherein: 1) the first plane and the second plane are parallel and spacedapart by a distance 2z; 2) all 8 coils are coaxial about an axis; 3) thefirst small coil encloses a first small filamentary loop with a radiusequal to 0.294922z, and the second small coil encloses a second smallfilamentary loop with a radius equal to 0.294922z; 4) the first mediumcoil encloses a first medium filamentary loop with a radius equal to0.590474z, and the second medium coil encloses a second mediumfilamentary loop with a radius equal to 0.590474z; 5) the first largecoil encloses a first large filamentary loop with a radius equal to1.031283z, and the second large coil encloses a second large filamentaryloop with a radius equal to 1.031283z; 6) the first extra large coilencloses a first extra large filamentary loop with a radius equal to2.632601z, and the second extra large coil encloses a second extra largefilamentary loop with a radius equal to 2.632601z; and 7) the firstfilamentary loops are coplanar with the first plane, and the secondfilamentary loops are coplanar with the second plane; whereby theelectromagnet provides a homogeneous magnetic field between the firstplane and second plane near the axis when energized with appropriatecurrents.
 19. The electromagnet of claim 18 further comprising a meansfor energizing the coils with electrical currents such that: a) thefirst small coil and the second small coil each carry Ampere-turns equalto an arbitrary value I_(s); b) the first medium coil and the secondmedium coil each carry Ampere-turns equal to 2.581089I_(s) to within 1%;c) the first large coil and the second large coil each carryAmpere-turns equal to 7.938104I_(s) to within 1%; and d) the first extralarge coil and the second extra large coil each carry Ampere-turns equalto 233.1785I_(s) to within 1%.
 20. The electromagnet of claim 18 furthercomprising a polarizing magnet for prepolarized magnetic resonanceimaging, wherein the polarizing magnet is disposed such that apolarizing magnetic field is provided in the region of the homogeneousmagnetic field.
 21. The electromagnet of claim 20 wherein the polarizingmagnet is a solenoidal magnet disposed between the first plane and thesecond plane such that the polarizing magnetic field is perpendicular tothe homogeneous magnetic field.
 22. The electromagnet of claim 20wherein the polarizing magnet is a solenoidal magnet disposed betweenthe first plane and the second plane such that the polarizing magneticfield is parallel to the homogeneous magnetic field.
 23. Theelectromagnet of claim 20 wherein the polarizing magnet is a solenoidalmagnet disposed outside the first plane and second plane and wherein thepolarizing magnet is coaxial with the 8 coils.
 24. The electromagnet ofclaim 18 wherein the coils are centered upon the correspondingfilamentary loops.
 25. The electromagnet of claim 18 wherein each coilhas cross-sectional dimensions less than 10% of the coil's radius.
 26. Abiplanar, symmetrical 10-coil electromagnet comprising: a) a first extrasmall coil b) a first small coil; c) a first medium coil; d) a firstlarge coil; e) a first extra large coil, wherein the first extra smallcoil, the first small coil, the first medium coil, the first large coil,and the first extra large coil are located in a first plane; f) a secondextra small coil; g) a second small coil; h) a second medium coil; i) asecond large coil; and j) a second extra large coil, wherein the secondextra small coil, the second small coil, the second medium coil, thesecond large coil, and the second extra large coil are located in asecond plane; wherein: 1) the first plane and the second plane areparallel and spaced apart by a distance 2z; 2) all 10 coils are coaxialabout an axis; 3) the first extra small coil encloses a first extrasmall filamentary loop with a radius equal to 0.236199z, and the secondextra small coil encloses a second extra small filamentary loop with aradius equal to 0.236199z; 4) the first small coil encloses a firstsmall filamentary loop with a radius equal to 0.457269z, and the secondsmall coil encloses a second small filamentary loop with a radius equalto 0.457269z; 5) the first medium coil encloses a first mediumfilamentary loop with a radius equal to 0.734885z, and the second mediumcoil encloses a second medium filamentary loop with a radius equal to0.734885z; 6) the first large coil encloses a first large filamentaryloop with a radius equal to 1.164685z, and the second large coilencloses a second large filamentary loop with a radius equal to1.164685z; 7) the first extra large coil encloses a first extra largefilamentary loop with a radius equal to 2.745650z, and the second extralarge coil encloses a second extra large filamentary loop with a radiusequal to 2.745650z; and 8) the first filamentary loops are coplanar withthe first plane, and the second filamentary loops are coplanar with thesecond plane; whereby the electromagnet provides a homogeneous magneticfield between the first plane and second plane near the axis whenenergized with appropriate currents.
 27. The electromagnet of claim 26further comprising a means for energizing the coils with electricalcurrents such that: a) the first extra small coil and the second extrasmall coil each carry Ampere-turns equal to an arbitrary value I_(s); b)the first medium coil and the second medium coil each carry Ampere-turnsequal to 2.255341I_(s) to within 1%; c) the first medium coil and thesecond medium coil each carry Ampere-turns equal to 4.938906I_(s) towithin 1%; d) the first large coil and the second large coil each carryAmpere-turns equal to 14.159399I_(s) to within 1%; and e) the firstextra large coil and the second extra large coil each carry Ampere-turnsequal to 386.32597I_(s) to within 1%.
 28. The electromagnet of claim 26further comprising a polarizing magnet for prepolarized magneticresonance imaging, wherein the polarizing magnet is disposed such that apolarizing magnetic field is provided in the region of the homogeneousmagnetic field.
 29. The apparatus of claim 28 wherein the polarizingmagnet is a solenoidal magnet disposed between the first plane and thesecond plane such that the polarizing magnetic field is perpendicular tothe homogeneous magnetic field.
 30. The apparatus of claim 25 whereinthe polarizing magnet is a solenoidal magnet disposed between the firstplane and the second plane such that the polarizing magnetic field isparallel to the homogeneous magnetic field.
 31. The apparatus of claim28 wherein the polarizing magnet is a solenoidal magnet disposed outsidethe first plane and second plane and wherein the polarizing magnet iscoaxial with the 10 coils.
 32. The electromagnet of claim 26 wherein thecoils are centered upon the corresponding filamentary loops.
 33. Theelectromagnet of claim 26 wherein each coil has cross-sectionaldimensions less than 10% of the coil's radius.